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The Complex Numbers are unfortunately named. Most people take the word complex to mean "difficult to understand", so the very name we give this family of numbers sets students up to think it's going to be a lot of hard work to understand them. This is sad, because they really are very very cool and not quite as difficult as people make them out to be.

It turns out, though, that the word complex has only recently attained the connotations of confusion.  The word complex according to my dictionary means simply "composed of multiple parts", which is plainly true of complex numbers: they have precisely two parts – a real part and an imaginary part.

My dictionary has another meaning for the word complex. The meaning above is the one used when we're using it as an adjective, but you can also use it as a noun. In that case a complex is an object which is composted of multiple parts. For example, a cinema complex is a building composed of multiple cinemas. In this sense, the phrase "complex number" is much more akin to "house boat" – a number which is a complex, like a boat which is a house. (I recently read this idea in "The Joy of x" by Steven Strogatz.)

I quite like both of these ideas. When people think of them as complicated numbers, it feels like they are making a value judgement, but with this older meaning of complex, it is value-free. It's a simple statement of fact about the structure of the numbers themselves. Sometimes you need to reclaim language to have a better perspective on the mathematical meaning.